.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
be non-zero real numbers. Define 
,
is equal to.
Y by Adventure10, Mango247
Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side 
. Prove that some three of these points are vertices of a triangle whose area is not greater than 
.
. Prove that some three of these points are vertices of a triangle whose area is not greater than 
.![[asy]
size(5cm);
path p = (2,0)--(0,0)--(2,2sqrt(3))--(4,0)--(2,0)--(1,sqrt(3))--(3,sqrt(3))--(2,0);
draw(p);
[/asy]](http://latex.artofproblemsolving.com/8/0/6/80638db8842c4e9cd1e15984f6dda4b27e0d01e0.png)
.
points in (or on the border of) one of the small triangles.The triangle formed by those 
points must have area less than or equal to 
